The $1/3 + \varepsilon$ is not a barrier anymore!
Resorting to estimates of exponential sums over subgroups due to Shkredov and Shteinikov, in the paper "On the congruence $x^{x} \equiv 1 \pmod{p}$" (PAMS, 144 (2016), no. 6, pp. 2411 - 2418), J. Cilleruelo (†) and M. Z. Garaev proved the following result:
Let $J(p)$ denote the number of solutions to the congruence $$x^{x} \equiv 1 \pmod{p}, \quad 1 \leq x \leq p-1.$$ Then, for any $\varepsilon>0$, there exists $c:=c(\varepsilon)>0$ such that $J(p) < c \, p^{27/82 + \varepsilon}$.
I am not totally sure, but I believe that this theorem is the state of the art on this problem...